
theorem
  for G being _Graph, P being Path of G holds P.length() c= G.order()
proof
  let G be _Graph, P be Path of G;
  A1: dom(P.vertexSeq() | P.length()) = Seg P.length() by Lm1;
  rng(P.vertexSeq() | P.length()) c= the_Vertices_of G;
  then card Seg P.length() c= card the_Vertices_of G by A1, CARD_1:10;
  then card P.length() c= card the_Vertices_of G by FINSEQ_1:55;
  hence thesis;
end;
